Sum of Integer Combinations is Integer Combination

From ProofWiki
Jump to navigation Jump to search


Let $a, b \in \Z$ be integers.

Let $S = \set {a x + b y: x, y \in \Z}$ be the set of integer combinations of $a$ and $b$.

Let $u \in S$ and $v \in S$.

Then $u + v \in S$.


As both $u, v \in S$, $u$ and $v$ can be expressed as:

\(\ds u\) \(=\) \(\ds a x_1 + b y_1\)
\(\ds v\) \(=\) \(\ds a x_2 + b y_2\)

where $x_1, x_2, y_1, y_2$ are integers.


\(\ds u + v\) \(=\) \(\ds a x_1 + b y_1 + a x_2 + b y_2\)
\(\ds \) \(=\) \(\ds a \paren {x_1 + x_2} + b \paren {y_1 + y_2}\)

As Integer Addition is Closed, both $x_1 + x_2$ and $y_1 + y_2$ are integers.

Hence the result.